We propose an extended version of the SEIRS modeled exploiting the Petri Net formalism to account for the population age distribution, that was classified into three groups: young individuals 0-19 years, adults 20-69 years, old adults aged at least 70 years. The following figure shows:
The population of the age class i is partitioned in the following seven compartments: (Si), (Ei), (Iu i), (Iq i), (Ih i), (Ri), (Di). With respect to the classical SEIRS model, we have added a transition from Iu i to Iq i to model the possibility to identify undetected cases and isolate them. In this way an individual in Iu i tested as positive to the SARS-CoV-2 swab will be moved in the quarantine regime, Iq i.
The force of infection adopted in the model is a time and age class dependent function and includes the following four terms:
A detailed description of the model (e.g., system of ordinary differential equations, parameters, etc) is reported in [1].
The calibration phase was performed to fit the model outcomes with the surveillance Piedmont infection and death data (from February 24st to May 2nd) using squared error estimator via trajectory matching. Hence, a global optimization algorithm, based on (Yang Xiang et al. 2012), was exploited to estimate 13 model parameters characterized by a high uncertainty due to their difficulty of being empirically measured:
Consistently, Figure 2A and 2B show that the calibrated model is able to mimic consistently the observed infected and death cases (red line respectively). In details, Figure 2A reports the cumulative trend of the infected individuals in which the undetected infected are showed in orange, the quarantine infected in light blue, and hospitalized infected in blue. The purple line reports the cumulative trend of the undetected cases diagnosed by SARS-CoV-2 swab tests. Differently Figure 2B shows the cumulative trend of deaths. In both histograms the surveillance data are reported as red line. Similarly, in Figure 3 the infected individuals for each age class are shown.
Fig.2) Number of (A) infected and (B) deceased individuals.
Fig.3) Number of infected individuals for each age class. The red curve represents the surveillance data, which does not account for undetected cases.
Three scenarios are implemented. In the the model is calibrated to fit the surveillance data (yellow). In the the model extends the second restriction beyond March, 21s t without implementing the third restriction (blue). In the the model consider a higher population compliance to the third governmental restriction (green).
Fig.4) Stochastic simulation results reported as traces (on the left) and as density distributions (on the right).
The daily evolution of infected individuals is shown varying on the columns the the efficacy of individual-level measures and on the rows the efficacy of community surveillance.
Fig.5) Pessimistic scenario in which the gradual reopening is not counterbalanced by any infection-control strategies
Fig.6) The daily evolution of infected individuals is shown varying on the columns the the efficacy of individual-level measures and on the rows the efficacy of community surveillance
Figure 5 shows the daily evolution of infected individuals computed by the stochastic simulation. The stacked bars report the undetected infected (orange), the quarantine infected (light blue), and hospitalized infected (blue). The red line shows the trend of the infected cases from surveillance data. The purple line reports the cumulative trend of the undetected cases diagnosed by SARS-CoV-2 swab tests. In Figure 6 we show the daily forecasts of the number of infected individuals with the efficacy of individual-level measures ranging from 0% to 60% on the columns (increasing by steps of 20%) and, on the rows, increasing capability (from 0% to 30%, by 10% steps) of identifying otherwise undetected infected individuals. These results are obtained as median value of 5000 traces for each scenario obtained from the stochastic simulation.
Pernice, S., M. Pennisi, G. Romano, A. Maglione, S. Cutrupi, F. Pappalardo, G. Balbo, M. Beccuti, F. Cordero, and R. A. Calogero. 2019. “A Computational Approach Based on the Colored Petri Net Formalism for Studying Multiple Sclerosis.” BMC Bioinformatics.
Yang Xiang, Sylvain Gubian, Brian Suomela, and Julia Hoeng. 2012. “Generalized Simulated Annealing for Efficient Global Optimization: The GenSA Package for R.” The R Journal. http://journal.r-project.org/.